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3 INTRODUCTION Chemical engineers frequently encounter the flow of a mixture of two fluids in Liquid-gas or liquid-vapor mixtures condensers and evaporators gas-liquid reactors combustion systems transport of some solid materials slurry of the solid particles in a liquid, and pumping the mixture through a pipe Liquid-liquid mixtures in emulsions as well as liquid-liquid extraction. Types of Flows

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4 Electrical Impedance Tomography (EIT) is a imaging modality in which the internal resistivity distribution is reconstructed based on the measured voltages on the surface object. COMPUTER Reconstruction Algorithm Reconstruction Algorithm Interface with Instrument Interface with Instrument V I Concept of electrical impedance tomography WHAT IS EIT?

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5 The forward problem calculates the voltages on the electrodes by using the injected current and assumed resistivity distribution. The inverse problem reconstructs the resistivity distribution by using the voltage measurements on the electrodes. Forward vs. inverse problem for EIT FORWARD SOLVER VS INVERSE SOLVER Inverse Solver Forward Solver An iterative inverse solver

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6 Governing Equation derived from Maxwell Equation Boundary Conditions: Complete Electrode Model MATHEMATICAL MODEL: FORWARD SOLVER Between electrodes, no current crosses the boundary if the impedance outside the imaged volume is much greater than that inside There is an existence of a thin, high-impedance layer beneath electrodes delivering current. This layer may be modelled as the limit of a thin layer of thickness d and impedance z/d as d goes to zero. (use ohms law) Beneath electrodes, neither potential nor the current crossing the boundary is known. Net current crossing the boundary beneath an electrode is equal to the current being delivered to it by tomograph electronics Constraints: For the solution to be unique

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17 Distinguishability can be defined as a measurement ability to differentiate between homogeneous and inhomogeneous conductivities inside the domain. Power distinguishability is defined as the measured power change between the homogeneous and inhomogeneous cases, divided by the power applied in homogeneous case. OPTIMAL CURRENT PATTERN (Front Points) 1. Trigonometric method with first 2 modes of cosine and sine (4 injections; 5 EKF states with repeated use of the first cosine) 2. Opposite method with e1-e9 and e5-e13 pairs (2 injections; 5 states with repeated use of e1-e9, e5-e13, e1-e9) 3. Cross method with e3-e7, e5-e13 pairs (2 injections; 5 states with repeated use of e3-e7, e5-e13, e3-e7) 4. Opposite method with e3-e11, e7-e15 pairs (2 injections; 5 states with repeated use of e3-e11, e7-e15, e3-e11) 5. Opposite method with e3-e11, e7-e15, e5-e13 pairs (3 injections; 5 states with repeated use of e3-e11, e7-e15). 1 2 3 4 5 1% Noise

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24 UNSCENTED KALMAN FILTER (1/4) Generate 2n+1 sigma points where n is the size of augmented vector Each point is the augmented vector Run the state equation Calculate predicted mean and covariance State Space Model:

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25 Run the measurement equation and find the mean Time update complete Create covariance matrices The sigma points should move towards the mean and at the same time, the sigma points on x domain should move towards the mean UNSCENTED KALMAN FILTER (2/4)

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34 APPENDIX: Derivation of Jacobian 1/10 In some cases, the voltages are measured only at some selected electrodes, not every electrode. Also, the selected electrodes may be different at each current pattern. The measured voltages at the measurement electrodes can be obtained as where, is the number of the measurement electrodes and is the measurement matrix. The element is set to 1 if the -th electrode is measured at the -th current pattern and otherwise set to zero. Furthermore, can be extracted directly from by introducing the extended mapping matrix and where. Therefore, we have where the extended measurement matrix is defined as If the pseudo-resistance matrix defined as or is given we can calculate the Jacobian matrix. The pseudo-resistance matrix can be easily obtained during the solution of the system equation or where and

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36 APPENDIX: Derivation of Jacobian 3/10 Since we are considering the stratified flow of two immiscible liquids therefore, the matrix B will be

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37 Assuming that the interface is represented by a set of linear piecewise interpolation functions:, unit pulse defined for Any small perturbation of results in small perturbation in and in where APPENDIX: Derivation of Jacobian 4/10

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38 APPENDIX: Derivation of Jacobian 5/10 Considering the interface for mesh crossing elements where For a small perturbation in only and will change The function can be expanded about the interface Finally, we have

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39 Five types of interface-crossing elements in case of an arbitrarily small perturbation of in. There are five types of interface-crossing elements when is perturbed by an arbitrarily small perturbation of. Assume that there are only two intersections of the interface and the mesh faces and the intersections and where. Recalling that the integration for each type will be evaluated as are denoted as is constant in a certain mesh, TYPE 1: TYPE 2: TYPE 3: TYPE 4: TYPE 5: APPENDIX: Derivation of Jacobian 6/10

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40 Fourier Coefficients Approach The derivative of the stiffness matrix with respect to the coefficient is APPENDIX: Derivation of Jacobian 7/10

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41 In order to obtain the Jacobian, now, let us consider the evaluation of the expression We define a new coordinate system where is the positively oriented coordinate along the closes curve, and is the coordinate outward normal from the region The perturbed boundary will be Therefore, APPENDIX: Derivation of Jacobian 8/10

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42 The Jacobian for the transformation of the coordinate will be The function can be expanded about the boundary We have APPENDIX: Derivation of Jacobian 9/10

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43 In this, is evaluated at the boundary. When differentiating with respect to, that is perturbing, we have and.On the other hand when differentiating with respect to, we have and. Finally, the derivative of the matrix with respect to the coefficients becomes where denotes the set of elements crossing If, and constant in each element, we have APPENDIX: Derivation of Jacobian 10/10